Difference between revisions of "Isotropic Material"
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| alpha || Thermal expansion coefficient || ppm/M || 40 | | alpha || Thermal expansion coefficient || ppm/M || 40 | ||
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The following properties are only allowed in MPM calculations: | |||
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! Property !! Description !! Units !! Default | |||
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| beta || Solvent expansion coefficient || 1/(wt fraction) || 0 | | beta || Solvent expansion coefficient || 1/(wt fraction) || 0 |
Revision as of 13:16, 1 April 2013
Constitutive Law
This material is a small strain, linear elastic material. The components of stress are related to components of strain by
[math]\displaystyle{ \sigma_{ij} = \bigl(\lambda\varepsilon_{ii} - 3K(\alpha \Delta T+\beta c)\bigr)\delta_{ij} + 2G\varepsilon_{ij} }[/math]
where λ is the Lame coefficient, K is bulk modulus, α is thermal expansion coefficient, ΔT is temperature difference, β is solvent expansion coefficient, c is solvent concentration, and G is shear modulus. Two other isotropic material properties are modulus, E, and Poisson's ratio, ν.
Material Properties
Although deformation properties of an isotropic material can be defined by any two of λ, K, G, E, and ν, this material's properties can only be defined by specifying any two (and exactly two) of E, G, and ν. Those three and other properties for isotropic materials are:
Property | Description | Units | Default |
---|---|---|---|
E | Tensile modulus | MPa | none |
G | Shear modulus | MPa | none |
nu | Poisson's ratio | none | none |
alpha | Thermal expansion coefficient | ppm/M | 40 |
The following properties are only allowed in MPM calculations:
Property | Description | Units | Default |
---|---|---|---|
beta | Solvent expansion coefficient | 1/(wt fraction) | 0 |
D | Solvent diffusion constant | mm2/sec | 0 |
kCond | Thermal conductivity | W/(m-K) | 0 |
(other) | Properties common all materials | varies | varies |
If you know K or λ instead of E, G, and ν, they are easily converted to E and ν. Given K and G:
[math]\displaystyle{ E = {9K \over 1+{3K\over G}} \qquad {\rm and} \qquad G = G }[/math]
or given λ and G:
[math]\displaystyle{ E = G\left({2 + {2G\over \lambda} \over 1 + {G\over \lambda}}\right) \qquad {\rm and} \qquad G = G }[/math]
or given K and ν:
[math]\displaystyle{ E = 3K(1-2\nu) \qquad {\rm and} \qquad \nu = \nu }[/math]
Other combinations are easily derived, but the above examples are the most common.
History Variables
None